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marithatkid
03.09.2020 •
Mathematics
What function do you know from calculus is such that its first derivative is itself?The above function is a solution of which of the following differential equations?a. y = ey.b. y = 1.c. y = y.d. y = y2.e. y = 2y.What function do you know from calculus is such that its first derivative is a constant multiple k of itself? The above function is a solution of which of the following differential equations?a. y = ky.b. y = yk.c. y = eky.d. y = k.e. y = y + k.
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Ответ:
a. y= e raise to power y
c. y = e^ky
Step-by-step explanation:
The first derivative is obtained by making the exponent the coefficient and decreasing the exponent by 1 . In simple form the first derivative of
x³ would be 2x³-² or 2x².
But when we take the first derivative of y= e raise to power y
we get y= e raise to power y. This is because the derivative of e raise to power is equal to e raise to power y.
On simplification
y= e^y
Applying ln to both sides
lny= ln (e^y)
lny= 1
Now we can apply chain rule to solve ln of y
lny = 1
1/y y~= 1
y`= y
therefore
derivative of e^y = e^y
The chain rule states that when we have a function having one variable and one exponent then we first take the derivative w.r.t to the exponent and then with respect to the function.
Similarly when we take the first derivative of y= e raise to power ky
we get y=k multiplied with e raise to power ky. This is because the derivative of e raise to a constant and power is equal to constant multiplied with e raise to power y.
On simplification
y= k e^ky
Applying ln to both sides
lny=k ln (e^y)
lny=ln k
Now we can apply chain rule to solve ln of y ( ln of constant would give a constant)
lny = ln k
1/y y~= k
y`=k y
therefore
derivative of e^ky = ke^ky
Ответ:
(4,2) is a part of (8,4)
Step-by-step explanation: